Confidence Interval

The construction follows a fixed template. Take the sample statistic as the centre, measure its sampling variability through the standard error, then scale that error by a reliability factor drawn from the appropriate sampling distribution. For a mean, the standard error is the sample standard deviation divided by the square root of the sample size; for a proportion, it is the square root of the estimated proportion times one minus that proportion, divided by the sample size, all under the root.

The reliability factor depends only on the confidence level and the distribution used. Under a standard normal distribution the two-sided factors are approximately 1.645 for 90 percent, 1.96 for 95 percent and 2.58 for 99 percent confidence. The product of the reliability factor and the standard error is the margin of error, and the interval runs one margin below and one margin above the point estimate.

The reliability factor comes from the z-distribution when the population variance is known, or when the sample is large enough that the central limit theorem makes the normal approximation acceptable. When the population variance is unknown and must be estimated from the sample, the correct factor comes from Student's t-distribution with degrees of freedom equal to the sample size minus one. Because the t-distribution has fatter tails, its factors exceed the corresponding z values, producing a wider interval that compensates for the extra uncertainty in the estimated variance.

As degrees of freedom increase, the t-distribution converges toward the normal, so the two approaches give nearly identical intervals in large samples. Using t when in doubt is the conservative choice: it never understates the width. Small samples drawn from clearly non-normal populations are the case where neither factor is reliable and a nonparametric or bootstrap approach is preferred.

The confidence level describes the long-run performance of the procedure, not the status of one realised interval. Repeat the sampling and construction indefinitely and 95 percent of the intervals produced will cover the fixed but unknown parameter. Once a specific interval has been computed, the parameter either lies inside it or does not; the randomness sat in the sample, not in the parameter.

This links directly to testing. A two-sided test at a 5 percent significance level rejects the null hypothesis precisely when the hypothesised value falls outside the 95 percent confidence interval. The interval therefore summarises the entire set of null values that would not be rejected, which is why regression tables can be read for significance simply by checking whether a coefficient interval straddles zero.

Confidence intervals appear throughout investment analysis: around estimated mean returns, around regression slope coefficients such as a beta estimate, around forecast values in a prediction interval, and around survey-based measures of sentiment or default frequency. Analysts also invert the construction to plan research, since specifying a target margin of error and confidence level determines the minimum sample size required, with the necessary size rising with the square of the desired precision.

At exam level a candidate should be able to identify the correct reliability factor, compute the standard error for a mean or a proportion, assemble the interval, state the width response to changes in confidence level, dispersion or sample size, and convert an interval into a reject or fail-to-reject conclusion for a two-sided test.